Is Non-Euclidean geometry really “Non”?

The definition of a straight line according to google.

I do not understand why I call these geometries "non-Euclidean". In my view, both hyperbolic and elliptical geometry are just a dimensional reference change of the plane, using the same elements described by Euclid. Both are described by curved planes, that is, analyzed three-dimensionaly. A straight line is no longer a straight line. Perhaps there is a lost axiom that has not been introduced to better define what a line is and not to confuse it with a curve. What I want to mean is that, all of these are the same elements but with another perspective. If we can define what a line really is, maybe we can debunk the axioms denying the parallels axiom. If anyone has understood my doubt, please tell me where I am wrong or if there is truth in my words. Thanks.

• In euclidean geometry the parallel postulate holds, in non-euclidean ones it doesn't. I doubt "dimensional reference change" and/or "analyzed three dimensionality" mean anything... The reason non-euclidean geometries are non-euclidean is extremely concrete and simple. – Mariano Suárez-Álvarez Apr 1 '18 at 4:47
• I suggest you read a modern exposition of euclidean geometry. Axioms do not define what a line is, for example —lines are undefined concepts, and that is in fact most of the point! – Mariano Suárez-Álvarez Apr 1 '18 at 4:49
• I think by "analyzed three dimensionally," he means that they have models which can be embedded in $\mathbb{R}^3$? – Lucky Apr 1 '18 at 4:50
• Where can I find such modern exposition, Mrs. @MarianoSuárez-Álvarez? – Nicolas Leskiu Apr 1 '18 at 4:51
• If Lucky's comment is a correct interpretation of your motivation, it's worth noting that the hyperbolic plane cannot in fact be isometrically embedded into $\mathbb{R}^3$; this is due to Hilbert. – Noah Schweber Apr 1 '18 at 5:29